Question

Question #eca0b

Answer 1

Do some factoring to get `3x^3-x^2+18x-6=(3x-1)(x^2+6)`

Explanation:

First we check to make sure the ratios are the same for consecutive terms. In plain English, this means we do this:
`color(blue)(3x^3)-color(blue)(x^2)+color(red)(18x)-color(red)(6)`
`->color(red)(6/(18x))=1/(3x)`
`->color(blue)(x^2/(3x^2))=1/(3x)`

Because the ratios are the same, we can factor by grouping.

Now, let's pull an `x^2` out of `3x^3-x^2`:
`3x^3-x^2+18x-6`
`->x^2(3x-1)+18x-6`

And a `6` out of `18x-6`:
`x^2(3x-1)+18x-6`
`->x^2(3x-1)+6(3x-1)`

Note that these have a common term of `(3x-1)`:
`x^2color(red)((3x-1))+6color(red)((3x-1))`

That means we can pull out a `3x-1` also:
`x^2color(red)((3x-1))+6color(red)((3x-1))`
`->color(red)((3x-1))(x^2+6)`

This last part may seem confusing. If it helps, replace `3x-1` with something less intimidating, like `a`:
`x^2a+6a`

For me, it's easier to see that we can pull out an `a` as a common factor:
`x^2a+6a`
`->a(x^2+6)`

Now just replace `a` with `3x-1`:
`(3x-1)(x^2+6)`